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<title>Chapter 8. Special Functions</title>
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<div class="chapter">
<div class="titlepage"><div><div><h1 class="title">
<a name="special"></a>Chapter 8. Special Functions</h1></div></div></div>
<div class="toc">
<p><b>Table of Contents</b></p>
<dl class="toc">
<dt><span class="section"><a href="math_toolkit/number_series.html">Number Series</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/number_series/bernoulli_numbers.html">Bernoulli
      Numbers</a></span></dt>
<dt><span class="section"><a href="math_toolkit/number_series/tangent_numbers.html">Tangent Numbers</a></span></dt>
<dt><span class="section"><a href="math_toolkit/number_series/primes.html">Prime Numbers</a></span></dt>
<dt><span class="section"><a href="math_toolkit/number_series/fibonacci_numbers.html">Fibonacci
      Numbers</a></span></dt>
</dl></dd>
<dt><span class="section"><a href="math_toolkit/sf_gamma.html">Gamma Functions</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/sf_gamma/tgamma.html">Gamma</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_gamma/lgamma.html">Log Gamma</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_gamma/digamma.html">Digamma</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_gamma/trigamma.html">Trigamma</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_gamma/polygamma.html">Polygamma</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_gamma/gamma_ratios.html">Ratios of Gamma Functions</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_gamma/igamma.html">Incomplete Gamma Functions</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_gamma/igamma_inv.html">Incomplete Gamma Function
      Inverses</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_gamma/gamma_derivatives.html">Derivative of
      the Incomplete Gamma Function</a></span></dt>
</dl></dd>
<dt><span class="section"><a href="math_toolkit/factorials.html">Factorials and Binomial Coefficients</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/factorials/sf_factorial.html">Factorial</a></span></dt>
<dt><span class="section"><a href="math_toolkit/factorials/sf_double_factorial.html">Double Factorial</a></span></dt>
<dt><span class="section"><a href="math_toolkit/factorials/sf_rising_factorial.html">Rising Factorial</a></span></dt>
<dt><span class="section"><a href="math_toolkit/factorials/sf_falling_factorial.html">Falling
      Factorial</a></span></dt>
<dt><span class="section"><a href="math_toolkit/factorials/sf_binomial.html">Binomial Coefficients</a></span></dt>
</dl></dd>
<dt><span class="section"><a href="math_toolkit/sf_beta.html">Beta Functions</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/sf_beta/beta_function.html">Beta</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_beta/ibeta_function.html">Incomplete Beta
      Functions</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_beta/ibeta_inv_function.html">The Incomplete
      Beta Function Inverses</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_beta/beta_derivative.html">Derivative of the
      Incomplete Beta Function</a></span></dt>
</dl></dd>
<dt><span class="section"><a href="math_toolkit/sf_erf.html">Error Functions</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/sf_erf/error_function.html">Error Function erf
      and complement erfc</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_erf/error_inv.html">Error Function Inverses</a></span></dt>
</dl></dd>
<dt><span class="section"><a href="math_toolkit/sf_poly.html">Polynomials</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/sf_poly/legendre.html">Legendre (and Associated)
      Polynomials</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_poly/legendre_stieltjes.html">Legendre-Stieltjes
      Polynomials</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_poly/laguerre.html">Laguerre (and Associated)
      Polynomials</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_poly/hermite.html">Hermite Polynomials</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_poly/chebyshev.html">Chebyshev Polynomials</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_poly/sph_harm.html">Spherical Harmonics</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_poly/cardinal_b_splines.html">Cardinal B-splines</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_poly/gegenbauer.html">Gegenbauer Polynomials</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sf_poly/jacobi.html">Jacobi Polynomials</a></span></dt>
</dl></dd>
<dt><span class="section"><a href="math_toolkit/bessel.html">Bessel Functions</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/bessel/bessel_over.html">Bessel Function Overview</a></span></dt>
<dt><span class="section"><a href="math_toolkit/bessel/bessel_first.html">Bessel Functions of
      the First and Second Kinds</a></span></dt>
<dt><span class="section"><a href="math_toolkit/bessel/bessel_root.html">Finding Zeros of Bessel
      Functions of the First and Second Kinds</a></span></dt>
<dt><span class="section"><a href="math_toolkit/bessel/mbessel.html">Modified Bessel Functions
      of the First and Second Kinds</a></span></dt>
<dt><span class="section"><a href="math_toolkit/bessel/sph_bessel.html">Spherical Bessel Functions
      of the First and Second Kinds</a></span></dt>
<dt><span class="section"><a href="math_toolkit/bessel/bessel_derivatives.html">Derivatives of
      the Bessel Functions</a></span></dt>
</dl></dd>
<dt><span class="section"><a href="math_toolkit/hankel.html">Hankel Functions</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/hankel/cyl_hankel.html">Cyclic Hankel Functions</a></span></dt>
<dt><span class="section"><a href="math_toolkit/hankel/sph_hankel.html">Spherical Hankel Functions</a></span></dt>
</dl></dd>
<dt><span class="section"><a href="math_toolkit/airy.html">Airy Functions</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/airy/ai.html">Airy Ai Function</a></span></dt>
<dt><span class="section"><a href="math_toolkit/airy/bi.html">Airy Bi Function</a></span></dt>
<dt><span class="section"><a href="math_toolkit/airy/aip.html">Airy Ai' Function</a></span></dt>
<dt><span class="section"><a href="math_toolkit/airy/bip.html">Airy Bi' Function</a></span></dt>
<dt><span class="section"><a href="math_toolkit/airy/airy_root.html">Finding Zeros of Airy Functions</a></span></dt>
</dl></dd>
<dt><span class="section"><a href="math_toolkit/ellint.html">Elliptic Integrals</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/ellint/ellint_intro.html">Elliptic Integral Overview</a></span></dt>
<dt><span class="section"><a href="math_toolkit/ellint/ellint_carlson.html">Elliptic Integrals
      - Carlson Form</a></span></dt>
<dt><span class="section"><a href="math_toolkit/ellint/ellint_1.html">Elliptic Integrals of the
      First Kind - Legendre Form</a></span></dt>
<dt><span class="section"><a href="math_toolkit/ellint/ellint_2.html">Elliptic Integrals of the
      Second Kind - Legendre Form</a></span></dt>
<dt><span class="section"><a href="math_toolkit/ellint/ellint_3.html">Elliptic Integrals of the
      Third Kind - Legendre Form</a></span></dt>
<dt><span class="section"><a href="math_toolkit/ellint/ellint_d.html">Elliptic Integral D - Legendre
      Form</a></span></dt>
<dt><span class="section"><a href="math_toolkit/ellint/jacobi_zeta.html">Jacobi Zeta Function</a></span></dt>
<dt><span class="section"><a href="math_toolkit/ellint/heuman_lambda.html">Heuman Lambda Function</a></span></dt>
</dl></dd>
<dt><span class="section"><a href="math_toolkit/jacobi.html">Jacobi Elliptic Functions</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/jacobi/jac_over.html">Overview of the Jacobi
      Elliptic Functions</a></span></dt>
<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_elliptic.html">Jacobi Elliptic
      SN, CN and DN</a></span></dt>
<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_cd.html">Jacobi Elliptic Function
      cd</a></span></dt>
<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_cn.html">Jacobi Elliptic Function
      cn</a></span></dt>
<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_cs.html">Jacobi Elliptic Function
      cs</a></span></dt>
<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_dc.html">Jacobi Elliptic Function
      dc</a></span></dt>
<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_dn.html">Jacobi Elliptic Function
      dn</a></span></dt>
<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_ds.html">Jacobi Elliptic Function
      ds</a></span></dt>
<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_nc.html">Jacobi Elliptic Function
      nc</a></span></dt>
<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_nd.html">Jacobi Elliptic Function
      nd</a></span></dt>
<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_ns.html">Jacobi Elliptic Function
      ns</a></span></dt>
<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_sc.html">Jacobi Elliptic Function
      sc</a></span></dt>
<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_sd.html">Jacobi Elliptic Function
      sd</a></span></dt>
<dt><span class="section"><a href="math_toolkit/jacobi/jacobi_sn.html">Jacobi Elliptic Function
      sn</a></span></dt>
</dl></dd>
<dt><span class="section"><a href="math_toolkit/jacobi_theta.html">Jacobi Theta Functions</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/jacobi_theta/jacobi_theta_overview.html">Overview
      of the Jacobi Theta Functions</a></span></dt>
<dt><span class="section"><a href="math_toolkit/jacobi_theta/jacobi_theta1.html">Jacobi Theta
      Function θ<sub>1</sub></a></span></dt>
<dt><span class="section"><a href="math_toolkit/jacobi_theta/jacobi_theta2.html">Jacobi Theta
      Function θ<sub>2</sub></a></span></dt>
<dt><span class="section"><a href="math_toolkit/jacobi_theta/jacobi_theta3.html">Jacobi Theta
      Function θ<sub>3</sub></a></span></dt>
<dt><span class="section"><a href="math_toolkit/jacobi_theta/jacobi_theta4.html">Jacobi Theta
      Function θ<sub>4</sub></a></span></dt>
</dl></dd>
<dt><span class="section"><a href="math_toolkit/lambert_w.html">Lambert <span class="emphasis"><em>W</em></span>
    function</a></span></dt>
<dt><span class="section"><a href="math_toolkit/zetas.html">Zeta Functions</a></span></dt>
<dd><dl><dt><span class="section"><a href="math_toolkit/zetas/zeta.html">Riemann Zeta Function</a></span></dt></dl></dd>
<dt><span class="section"><a href="math_toolkit/expint.html">Exponential Integrals</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/expint/expint_n.html">Exponential Integral En</a></span></dt>
<dt><span class="section"><a href="math_toolkit/expint/expint_i.html">Exponential Integral Ei</a></span></dt>
</dl></dd>
<dt><span class="section"><a href="math_toolkit/hypergeometric.html">Hypergeometric Functions</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_1f0.html">Hypergeometric
      <sub>1</sub><span class="emphasis"><em>F</em></span><sub>0</sub> </a></span></dt>
<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_0f1.html">Hypergeometric
      <sub>0</sub><span class="emphasis"><em>F</em></span><sub>1</sub> </a></span></dt>
<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_2f0.html">Hypergeometric
      <sub>2</sub><span class="emphasis"><em>F</em></span><sub>0</sub></a></span></dt>
<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_1f1.html">Hypergeometric
      <sub>1</sub><span class="emphasis"><em>F</em></span><sub>1</sub></a></span></dt>
<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_pfq.html">Hypergeometric
      <sub>p</sub>F<sub>q</sub></a></span></dt>
<dt><span class="section"><a href="math_toolkit/hypergeometric/hypergeometric_refs.html">Hypergeometric
      References</a></span></dt>
</dl></dd>
<dt><span class="section"><a href="math_toolkit/powers.html">Basic Functions</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/powers/sin_pi.html">sin_pi</a></span></dt>
<dt><span class="section"><a href="math_toolkit/powers/cos_pi.html">cos_pi</a></span></dt>
<dt><span class="section"><a href="math_toolkit/powers/log1p.html">log1p</a></span></dt>
<dt><span class="section"><a href="math_toolkit/powers/expm1.html">expm1</a></span></dt>
<dt><span class="section"><a href="math_toolkit/powers/cbrt.html">cbrt</a></span></dt>
<dt><span class="section"><a href="math_toolkit/powers/sqrt1pm1.html">sqrt1pm1</a></span></dt>
<dt><span class="section"><a href="math_toolkit/powers/powm1.html">powm1</a></span></dt>
<dt><span class="section"><a href="math_toolkit/powers/hypot.html">hypot</a></span></dt>
<dt><span class="section"><a href="math_toolkit/powers/ct_pow.html">Compile Time Power of a Runtime
      Base</a></span></dt>
<dt><span class="section"><a href="math_toolkit/powers/rsqrt.html">Reciprocal square root</a></span></dt>
<dt><span class="section"><a href="math_toolkit/powers/logaddexp.html">logaddexp and logsumexp</a></span></dt>
</dl></dd>
<dt><span class="section"><a href="math_toolkit/sinc.html">Sinus Cardinal and Hyperbolic Sinus
    Cardinal Functions</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/sinc/sinc_overview.html">Sinus Cardinal and Hyperbolic
      Sinus Cardinal Functions Overview</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sinc/sinc_pi.html">sinc_pi</a></span></dt>
<dt><span class="section"><a href="math_toolkit/sinc/sinhc_pi.html">sinhc_pi</a></span></dt>
</dl></dd>
<dt><span class="section"><a href="math_toolkit/inv_hyper.html">Inverse Hyperbolic Functions</a></span></dt>
<dd><dl>
<dt><span class="section"><a href="math_toolkit/inv_hyper/inv_hyper_over.html">Inverse Hyperbolic
      Functions Overview</a></span></dt>
<dt><span class="section"><a href="math_toolkit/inv_hyper/acosh.html">acosh</a></span></dt>
<dt><span class="section"><a href="math_toolkit/inv_hyper/asinh.html">asinh</a></span></dt>
<dt><span class="section"><a href="math_toolkit/inv_hyper/atanh.html">atanh</a></span></dt>
</dl></dd>
<dt><span class="section"><a href="math_toolkit/owens_t.html">Owen's T function</a></span></dt>
<dt><span class="section"><a href="math_toolkit/daubechies.html">Daubechies Wavelets and Scaling
    Functions</a></span></dt>
<dt><span class="section"><a href="math_toolkit/ccmath.html">Constexpr CMath</a></span></dt>
</dl>
</div>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
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